2 Dissipative particle dynamics with energy conservation: Heat conduction that is dissipated due to the velocity dependent forces is transformed into

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1 International Journal of Modern Physics C, fc World Scientic Publishing Company DISSIPATIVE PARTICLE DYNAMICS WITH ENERGY CONSERVATION: HEAT CONDUCTION M. RIPOLL, P. ESPA ~NOL Departamento de Fsica Fundamental, UNED, Apartado 604 Madrid 28080, Spain and M.H. ERNST Institute for Theoretical Physics, Utrecht University, Princetonplein 5, P.O. Box TA Utrecht, The Netherlands Received (received date) Revised (revised date) We study by means of numerical simulations the model of dissipative particle dynamics with energy conservation for the simple case of thermal conduction. It is shown that the model displays correct equilibrium uctuations and reproduces Fourier law. The connection between \mesoscopic coarse-graining" and \resolution" is claried.. Introduction Mesoscopic approaches to the study of complex uids are receiving a great deal of attention in view of their potential to address problems with disparate time-scales. The method of dissipative particle dynamics introduced originally as an hybrid between lattice gas and molecular dynamics simulations, has proven to be a exible and powerful tool in the study of complex uids as colloidal suspensions 2;;4, porous ow, polymer suspensions 5, and multicomponent ows 6. DPD models a uid as a collection of point particles that are interpreted as representing the center of mass of a mesoscopic cluster of the atoms or molecules that constitute the uid. By formulating physically motivated interactions (that is, dissipative forces that conserve momentum ;7 ) between these \lumps" of uid, it is assured that the macroscopic behavior of the system is hydrodynamic 8;9. Most importantly, a random interaction is introduced in order to account for thermal uctuations that occur at the mesoscopic level and are responsible for Brownian eects appearing in complex uids. One of the drawbacks of classic DPD is that the total energy of the system is not conserved because the forces are velocity dependent. This has been recently remedied simultaneously and independently in Refs. 0;, where an internal energy variable and a temperature is introduced for each particle. The mechanical energy pep@sfun.uned.es

2 2 Dissipative particle dynamics with energy conservation: Heat conduction that is dissipated due to the velocity dependent forces is transformed into internal energy of the particles. This viscous heating process is accompanied by a thermal conduction process where exchange of internal energy between neighboring particles occurs when dierences of temperatures exist. The new DPD algorithm with energy conservation opens up the possibility of studying not only rheological aspects of complex uids but also thermal issues. It is therefore important to validate the technique in simple situations for which the dynamical aspects are well-known, either analytically or by means of other numerical simulations techniques. As a rst step towards the understanding of the behavior of DPD with energy conservation we focus in this paper on the simplest problem of conduction in a quiescent uid (or a solid). This is a particular case of the model introduced in Refs. 0;. The particles are at rest and located randomly and represent portions of material that can interchange energy. This simple model can be regarded as a way of solving a uctuating heat conduction equation. In section 2 we review the model that was introduced in Refs. 0;, for the case that only heat conduction takes place. In section we further specify the model functions. In section 4 we present simulation results that show the validity of the simulation method by comparing the numerical results with the theoretical results. Finally, we end up with a section of conclusions. 2. The Model We have introduced in Ref. 0 a mesoscopic model for describing a uctuating viscous and thermally conducting uid. Here we apply this model to a quiescent uid (or solid) within a box. The system is represented by a set of N particles of xed positions r i which are distributed at random within the box. These particles are understood as thermodynamic subsystems of the whole system, that is, small portions of material that have a suciently large number of degrees of freedom. To each particle we associate an internal energy variable i an entropy function s( i ), and a temperature T i? i =@ i. Each particle represents a mesoscopic portion of the material and its internal energy content is subject to thermal uctuations, which in a continuum theory would be described by a random heat ux. 2 Dierences in temperatures between neighboring particles will cause a transfer of internal energy. Therefore, the following stochastic dierential equation for the internal energy of each particle is postulated d i = X j!(r ij ) ij T i? T j dt + X j (2k B!(r ij ) ij ) =2 dw ij () The rst term in the r.h.s. is deterministic and species that a temperature difference causes exchange of energy. The second term is stochastic and takes into account thermally induced uctuations in each particle caused by a random interchange of energy between particles. The notation is as follows. k B is Boltzmann's constant, r ij = jr i? r j j is the distance between particles i; j and the dimensionless weight function!(r) determines the range of inuence between particles. The pa-

3 Dissipative particle dynamics with energy conservation: Heat conduction rameter ij governs the overall amplitude of the deterministic and random parts. It depends in general on the state variables of particles i; j and is symmetric under particle interchange. The random \heat ux" from particle i to j is expressed in terms of the increments of the Wiener process dwij. The increments of the Wiener process are antisymmetric under particle interchange dwij =?dw ji in such a way that the total internal energy of the system governed by Eqn. () is exactly conserved d dt i i = 0. Because ij might depend in general on i ; j we need to P provide a stochastic interpretation of Eqn. (), as the noise is multiplicative. We choose It^o interpretation. It has been shown that, ij ij = 0 y, then Eqn. () is mathematically equivalent to a Fokker-Planck equation which has as equilibrium solution eq () / exp ( k? B X i s( i ) ) P( X k k ) (2) P where P( k k) is a function of the total internal energy which is determined by the initial distribution of total energy. Particular examples are given by the canonical and microcanonical ensembles mic ( ; : : : ; N ) = can ( ; : : : ; N ) = ( (E 0 ; N) exp ( Z(; N) exp k? B k? B X i X i s( i ) ) ( X i s( i )? i ) i? E 0 ) Here, the factors (E 0 ; N) and Z(; N) are obtained by normalization.. Perfect solid and Fourier law In this section we further specify the model by providing the undetermined functions T (), ( + 0 ) and!(r) appearing in Eqn. (). The simplest possible form for the equation of state T () is that of a perfect solid, which is a very good approximation for metals. The perfect solid has the following equation of state T () = () C v (4) where the heat capacity of the mesoscopic particles C v is a constant independent of the energy. It is an extensive property that depends on the \size" of the particles. For mesoscopic particles, the dimensionless heat capacity C v =k B. The entropy s() is obtained by integrating the temperature and, apart from irrelevant constant additive terms, it is given by s() = C v ln. Now, let us assume the following functional form for ij, y This implies that if ij depends only on i ; j then, necessarily, it will be a function of i + j. This can be proved by considering a Taylor expansion of a function of two variables and using the fact that, at each point, both partial derivatives coincide.

4 4 Dissipative particle dynamics with energy conservation: Heat conduction ij = C v~ 2 T 2 i + j = C v~ 2 4 (T 2 i + T j ) 2 (5) where is the average distance between particles (this is = n?= and n is the average density number) and ~ has dimensions of a diusion coecient. In the last equality, Eqn. (4) has been used. Substitution of Eqn. (5) into the dynamical equation for the energy leads to the following equation for the temperature of a particle 2kB ~ dt i = ~ 2 X j!(r ij ) (T i + T j ) 2 4T i T j (T j? T i )dt + C v 2 =2 X j! =2 (r ij ) T i + T j dwij 2 (6) (Ti+Tj )2 The factor 4T it j should be very close to if the temperature dierences between particles within an action sphere are small. We see, then, that for the perfect solid by a proper selection of ij the form of the postulated SDE for the energy leads to a discrete version of the Fourier law of heat conduction. Note that the factor 2 in the rst term of the left hand side of Eqn. (6) can be understood as the \lattice spacing" squared that would arise in a nite dierence discretization of the Laplacian in the heat conduction equation. In a sense, Eqn. (6) can be understood as a nite dierence discretization of the macroscopic heat equation on a random lattice which, in addition, has the correct form for the thermal uctuations. We will assume that the weight function!(r) has the form of the Lucy weight function, which is a smooth bell-shaped function with continuous derivatives. It is given by!(r) = 05 6s + r r c? r r c (7) where r c is range of interaction between particle and s = r c = is the overlapping coecient which gives, essentially, the number of neighbors of a given particle. We have normalized the weight function as Z d d r!(r) = n (8) 4. Simulation results In this section we present some results obtained from numerical simulation of Eqn. (6). The physical system is assumed to be in a three dimensional cubic box of edge length L. Initially the box is seeded with N mesoscopic particles located at random. The density is then n = N=L and the typical interparticle distance is = n?=. Periodic boundary conditions are assumed in the y; z directions and in the x direction we assume either periodic boundary conditions when considering equilibrium situations or we impose the temperature at x =?L=2 and x = L=2. This is achieved by considering two extra layers of particles in each boundary in the

5 Dissipative particle dynamics with energy conservation: Heat conduction 5 x direction lled with particles that are at a constant temperature. These layers act, therefore, as thermal baths. They have a width as large as r c in order that any particle within the system interacts with the same number of particles. The basic model parameters are the following ones: ~, N, L, r c, C v, T 0. Here, T 0 is a reference temperature, that of equilibrium or, in the case of nonequilibrium situations, that of the colder thermal bath. We work in reduced units such that L is the unit of length, L 2 =~ is unit of time, T 0 is the unit of temperature, and k B is taken as the unit of entropy or heat capacity. We introduce the dimensionless heat capacity of mesoscopic particles = C v =k B and the dimensionless heat capacity of the total simulated sample, which is C L = N because heat capacity is an extensive property. On the other hand, C L = L c v =k B, where c v is the heat capacity per unit volume, which is a material constant. For example, c v = : J=Km for copper at room temperature. Therefore, by xing C L we are xing the actual volume of the sample. For copper, a value :60?0 m 0:. In most of the simulations we use C L = 0 8 and this xes the value of depending on the number of particles in the system. Note that by xing C L, large N implies small, in accordance with the idea that the heat capacity of the mesoscopic particles is proportional to its typical size. We note that the overall intensity of the noise in (6) is proportional to (k B =C v ) =2 =?=2 and, therefore, inversely proportional to the square root of the volume of the particles. 4 The higher the number density of mesoscopic particles used to discretize a given sample, the smaller is the size of such particles and, therefore, the larger is the thermal noise. Eqns. (6) are solved with a conventional Euler method. For values of < 0, the stochastic term produces from time to time a temperature which is negative for some particle. This has disastrous consequences because the factor accompanying (T i? T j ) in the deterministic term of (6) is negative and produces a ow of energy from the cold particles to the hot particles. When this happens the system becomes unstable. However, this never occurs for suciently large values of as the ones we typically use and no instabilities are observed. In order to check that the code performs properly, we rst run a simulation with fully periodic boundary conditions. Initially, all the particles are assumed to have the same temperature T =. After some equilibration period the system reaches equilibrium. The energy is exactly conserved (no round-o errors) because we use a Verlet list 5, in such a way that what is gained by a particle is exactly what is lost by the rest of particles. In Fig. we show the probability density distribution of the energy of a particle as obtained from simulation of Eqn. () for two dierent values of = C v =k B, the dimensionless heat capacity of the mesoscopic particles, for a system of N = 000. The larger the larger the average value of the energy and variance. However, the relative uctuations are smaller, proportional to N?=2. Also shown in Fig. are the canonical f can () and microcanonical f mic () theoretical results for the given values of. The analytical expressions can be obtained from () and are given by C L = 0 8 implies a submicron sample size of L = C = L

6 6 Dissipative particle dynamics with energy conservation: Heat conduction f can () = f mic () = M?( + ) () expf?g E 0 E 0? E 0 (N?)(+)? (9) where?(x) is the gamma function. The parameter can be explicitly computed for a perfect solid by requiring that the average energy of a particle is E 0 =N, where E 0 is the total energy of the isolated system. The result is = N( + )=E 0. In the thermodynamic limit the canonical and microcanonical distributions are identical. In practice, they are indistinguishable for values N > 50. The perfect coincidence of the theoretical and simulation results in Fig. gives condence on the implementation of the numerical code used f() Fig.. The probability density distribution f() of the energy of a given particle. Two simulations with = 50 (left) and = 00 (right) are presented. Other parameters are N = 000, s = :5, T eq =. Also shown, although not visible because they are perfectly superimposed, are the theoretical predictions for the microcanonical and canonical distributions for the given values of. Next, we compute the thermal diusivity by a macroscopic measurement. Two heat baths are constructed with temperatures T cold and T hot in such a way that a gradient (T hot? T cold )=L is applied. A steady state is reached for suciently long times. Fig. 2 shows the average of the temperature once the steady state has been reached for dierent number of particles for the case that T cold =, T hot = 2 and the overlapping coecient is s = :5. For N = 00 the spatial noise is considerable. Fluctuations in this graph are not due to statistical noise but to the

7 Dissipative particle dynamics with energy conservation: Heat conduction 7 spatial inhomogeneities that appear for small number of particles and they would smear out by averaging over dierent realizations of the initial spatial conguration of the particles. For larger number of particles a smooth linear prole is achieved, in accordance with our expectation that the discrete model reproduces Fourier law. The temperature eld T (x) is obtained by binning the x axis and averaging the temperatures of the particles within each bin of volume xl T (x) x Fig. 2. Average temperature of the particles in each of the 25 bins in which the x axis of the box has been divided. Solid line is the theoretical Fourier prole. Open circles correspond to N = 00 and solid circles correspond to N = 000 particles within the box. Other parameters are s = :5, C L = 0 8. We have performed a series of simulations with dierent temperature gradients and have computed the macroscopic heat ux in the steady state. Two dierent ways of computing the macroscopic heat ux have been used. In the rst case, the macroscopic heat ux is obtained by computing at each time step the energy gained by the cold bath (which is equal to the energy lost by the hot bath) and dividing it by the time step, this is E=t. The second way follows from the microscopic denition of the heat ux as obtained from projection operator methods or from kinetic theory 6 X Q (0) ij!(r ij ) ij T i? T j rij 2 All transport uxes are caused by collisional transfer. As the particles do no move, there are no kinetic uxes. This second form produces better results because it involves a very large number of pairs of particles and it is therefore adopted here. In Fig. we plot the macroscopic heat ux in terms of the applied temperature gradient. A linear dependence is obtained, which allows to extract the thermal

8 8 Dissipative particle dynamics with energy conservation: Heat conduction diusivity from the slope Q x (rt ) x Fig.. Average x component of the heat ux as a function of the imposed temperature gradient. A linear dependence is obtained with small deviations at large gradients. The slope of the straight line gives the thermal diusivity. When this experiment is performed for dierent number densities and dierent overlapping we can compare with the results from kinetic theory as developed in Ref. 6. Fig. 4 shows the value of the macroscopic thermal diusivity as a function of the overlapping s. Also shown is the kinetic theory prediction for the thermal diusivity 6, reading, D = s2 ~ () 24 We observe that the theoretical prediction improves for large overlapping. The kinetic theory result () has been obtained through the choices Eqns. (4) and (5) and shows no dependence on the density. This is a consequence of introducing the factor?2 in Eqn. (5). We verify in Fig. 5 that simulations performed at dierent number density, with the remaining parameters s; C L held xed, show that the thermal diusivity D is indeed independent of n, as predicted in Eqn. (). 5. Conclusions In this paper we have applied the model introduced in Ref. 0 to the case of heat conduction in a random solid. We have chosen the equation of state s( i ) for a perfect solid model and a particular model for ij, Eqn. (5). It has been shown that the model has the correct equilibrium properties and that it reproduces Fourier's law in non-equilibrium situations. We have also corroborated, by means

9 Dissipative particle dynamics with energy conservation: Heat conduction D Fig. 4. Thermal diusivity coecient for dierent values of the overlapping. Also shown (solid line) is the kinetic theory prediction. s 2 of simulations, the kinetic theory predictions for this model which are presented elsewhere. Kinetic theory predicts a thermal diusivity which depends only on the overlapping and not on the density. By introducing the factor?2 in the denition of ij we have shown that it is possible to interpret n not as the density but as the resolution at which the physical problem is being studied. A similar discussion about resolution was presented for the case of DPD with no energy conservation in Ref. 7. Acknowledgments M.R. acknowledges the support of a F.P.I. grant from Ministerio de Educacion. P.E. has received partial support from DGICYT Project No PB and by E.C. Contract ERB-CHRXCT References. P.J. Hoogerbrugge and J.M.V.A. Koelman, Europhys. Lett. 9, 55 (992). 2. J.M.V.A. Koelman and P.J. Hoogerbrugge, Europhys. Lett. 2, 69 (99).. E.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, and P. van der Schoot, Phys. Rev. E 55, 24 (997). 4. E.S. Boek, P.V. Coveney, and H.N.W. Lekkerkerker, J. Phys.: Condens. Matter 8, 9509 (997). 5. A.G. Schlijper, P.J. Hoogerbrugge, and C.W. Manke, J. Rheol. 9, 567 (995). 6. P.V. Coveney and K. Novik, Phys. Rev. E 54, 54 (996). 7. P. Espa~nol and P. Warren, Europhys. Lett. 0, 9 (995).

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